Depth of a module

One of the cohomological characteristics of a module over a commutative ring. Let $ A $ be a Noetherian ring, let $ I $ be an ideal in it and let $ M $ be an $ A $- module of finite type. Then the $ I $- depth of the module $ M $ is the least integer $ n $ for which

$$ \mathop{\rm Ext} _ {A} ^ {n} ( A/I, M) \neq 0. $$

The depth of a module is denoted by $ \mathop{\rm depth} _ {I} ( M) $ or by $ \mathop{\rm prof} _ {I} ( M) $. A different definition can be given in terms of an $ M $- regular sequence, i.e. a sequence of elements $ a _ {1} \dots a _ {k} $ of $ A $ such that $ a _ {i} $ is not a zero divisor in the module

$$ M/( a _ {1} \dots a _ {i - 1 } ) M. $$

The $ I $- depth of $ M $ is equal to the length of the largest $ M $- regular sequence consisting of elements of $ I $. The maximal ideal is usually taken for $ I $ in the case of a local ring $ A $. The following formula is valid:

$$ \mathop{\rm prof} _ {I} ( M) = \ \inf _ {\mathfrak p \supset I } \ ( \mathop{\rm prof} ( M _ {\mathfrak p} )), $$

where $ \mathfrak p $ denotes a prime ideal in $ A $, while $ M _ {\mathfrak p} $ is considered as a module over the local ring $ A _ {\mathfrak p} $.

The concept of the depth of a module was introduced in [1] under the name of homological codimension. If the projective dimension $ \mathop{\rm dh} ( M) $ of a module $ M $ over a local ring $ A $ is finite, then

$$ \mathop{\rm dh} ( M) + \mathop{\rm prof} ( M) = \mathop{\rm prof} ( A). $$

In general $ \mathop{\rm prof} ( M) $ is not larger than the dimension of $ M $.

The depth of a module is one of the basic tools in the study of modules. Thus, Cohen–Macaulay modules and rings (cf. Cohen–Macaulay ring) have been defined in terms of the depth of modules. The Serre criterion ( $ S _ {k} $) for an $ A $- module $ M $:

$$ \mathop{\rm prof} M _ {\mathfrak p} \geq \inf \ ( k, \mathop{\rm dim} M _ {\mathfrak p} ) $$

for all prime ideals $ \mathfrak p $ in $ A $, often proves to be useful. Finally, the depth of modules is closely connected with local cohomology modules: The statement

$$ \mathop{\rm prof} _ {I} ( M) \geq n $$

is equivalent to saying that the local cohomology modules $ H _ {I} ^ {i} ( M) $ vanish if $ i < n $.

References

[1]	M. Auslander, D.A. Buchsbaum, "Homological dimension in Noetherian rings" Proc. Nat. Acad. Sci. USA , 42 (1956) pp. 36–38
[2]	J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965)
[3]	A. Grothendieck, "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , SGA 2 , IHES (1965)